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Accelerating Ant Colony Optimization By
Using Local Search
Nabila Tabassum ID: 11301006 Maruful Haque ID:11201003 A DISSERTATION Submitted in partial fulfillment of the requirements for the degree of Bachelor of Science in Computer Science and Engineering Supervisor:Dilruba Showkat
August 2015
Contact Information:
Author 01:
Nabila Tabassum
E-mail: [email protected]
Author 02:
MarufulHaque
E-mail: [email protected]
Supervisor:
DilrubaShowkat
E-mail: [email protected] Lecturer
Department of Computer Science and Engineering
Brac University
Dhaka
Table of Contents
Acknowledgement ………………………………………………………………01
Abstract…………………………………………………………………………..02
CHAPTER 01
Introduction
03-04
1.1 ACO is member of swarm intelligence………………………………….....05
1.2 Motivation………………………………………………………………......05
1.3 Research Questions……………………………………………………........05
CHAPTER 02
Background
2.1 Optimization……………………………………………………………..........06
2.1.1 Constrained Optimization……………………………………………..........06
2.1.2 Unconstrained Optimization…………………………………………......06
2.1.3 Dynamic Optimization……………………………………………….......07
2.2.1 Single Object Optimization…………………………………………...07-10
2.2.2 Multiple Object Optimization…………………………………………....10
2.3.1 Evolutionary Algorithm………………………………………………......11
2.3.2 Population &Selection………………………………………………….11-12
2.4.1 Swarm Intelligence…………………………………………………….12-13
2.3.3 Crossover &Mutation……………………………………………….…...13
CHAPTER 3
3.1The Basic Model of ACO algorithm…………………………………......….15
3.1.1 Generate solution………………………………………………………........16
3.1.2 Daemon Action……………………………………………………….......16
3.1.3 Pheromone Update……………………………………………………..16-18
3.2 SBX ACO………………………………………………………………..19-22
3.3Advantage and disadvantage of ACO…………………………………….22-23
3.4 Recent Research and Advanced Topic on ACO…………………………...23
CHAPTER 4
Bench mark Function…………………………………………………………24-29
CHAPTER 5
5.1 Performance evaluation Criteria………………………………………........30
5.1.1 Convergence…………………………………………………………........30
5.1.2Experimental Setup…………………………………………………......…31
5.1.3 Comparison with ACO……………………………………………….......31
5.1.4 Comparison With DE and DEachSPX………………………………..31-35
CHAPTER 6
Application…………………………………………………………………....35-40
CHAPTER 7
Conclusion…………………………………………………………………...........41
Future plan………………………………………………………………........…..42
LIST OF FLOW CHART :
2.1 flowchart of ABC………………………………………………………...........8
2.2 flowchart of PSO………………………………………………………............9
3.2 flowchart of SBX and polynomial……………………………………........…18
3.3flowchart of SBX-ACO………………………………………………….........20
LIST OF FIGURE:
3.1 ACO model……………………………………………………………...........14
4.1 Sphere Function………………………………………………………………25
4.2 Rosenbrock Function…………………………………………………………26
4.3 Scaffer Function………………………………………………………………27
4.4 Griewank Function……………………………………………………………28
4.5 Ackley Function………………………………………………….......……….30
5.1.1-5.1.5 Graphical explanation……………………………………….....….….32
LIST OF TABLE:
5.1.1………………………………………………………….……………………31
5.1.2……………………………………………………………………………….32
1 Acknowledgement
First and foremost, we would like to express our gratitude to Almighty Allah (SWT) who gave me the opportunity, determination, strength and intelligence to complete my thesis. We want to acknowledge our fellow classmates who have consistently support us throughout the thesis.We would like to thank our supervisor Dilruba Showkat sincerely for her consistent supervision, guidance and unflinching encouragement in accomplishing our work and help us to present our idea properly. 2 Abstract
Optimization is very important fact in terms of taking decision in mathematics, statistics,
computer science and real life problem solving or decision making application. Many different
optimization techniques have been developed for solving such functional problem. In order to
solving various problem computer Science introduce evolutionary optimization algorithm and
their hybrid. In recent years, test functions are using to validate new optimization algorithms and
to compare the performance with other existing algorithm. There are many Single Object
Optimization algorithm proposed earlier. For example: ACO, PSO, ABC. ACO is a popular
optimization technique for solving hard combination mathematical optimization problem. In this
paper, we run ACO upon five benchmark function and modified the parameter of ACO in order
to perform SBX crossover and polynomial mutation. The proposed algorithm SBXACO is tested
upon some benchmark function under both static and dynamic to evaluate performances. We
choose wide range of benchmark function and compare results with existing DE and its hybrid
DEahcSPX from other literature are also presented here.
3 CHAPTER 1
Introduction
Science, invention, applied mathematics, economic analysis, industrial, technological and
managerial decisions making need solution in an optimal way. In this modern era, day by day the
world becomes more first at the same time more and more complex and competitive so that
decision making must be taken in an optimal way for better results in a faster way. Therefore
optimization is very important and concerning act of obtaining the best result under given
situations. This is the reason behind why optimization has been a popular research topic for
decades. Optimization originated in the 1940s, when the British military faced the problem of
allocating limited resources (for example fighter airplanes, submarines and so on) to several
activities [6]. In computer science over the decades, several researchers have generated different
solutions to linear and non-liner optimization problems. Among them evolutionary algorithms is
most popular and interesting part of modern computer science. Evolutionary algorithm are
biology-inspired soft computing methods have been widely used in different optimization
problem solving cases. Mathematically an optimization problem has a fitness function,
describing the problem under a set of constraints which represents the solution space for the
problem [1]. Especially, considering dynamic optimization problems it is quite interesting how
their objective functions changed over time, which causes changes in the position of optima as
well as the characteristics of the search space. This leads to the fact where existing optima may
disappear, while new optima may appear. Optimization under the dynamic environments is a
challenging task that attracts great attention [8].However, most of the traditional optimization
techniques have calculated the first derivatives to locate the optima on a given constrained
surface but modern optimization methods which are known as nontraditional optimization works
dynamically for constrained and unconstrained problems. Due to the difficulties in evaluation the
first derivative for many rough and discontinuous optimization spaces, several derivatives free
optimization methods have been constructed in recent time [15]. There are many single
optimization algorithm have been introduced in last few years. But there is no known single
optimization method available for solving all kind of optimization problems. In order to develop
the efficiency a lot of hybrid single optimization problem have been developed for solving
different types of optimization problems more efficiently. It becomes very popular and useful in
terms of research now days. The modern single optimization methods (sometimes called
nontraditional optimization methods) are very powerful. These are particle swarm optimization
algorithm, neural networks, genetic algorithms, ant colony optimization, differential evolution,
artificial immune systems, and fuzzy optimization, the Clonal Selection Algorithm (CSA), an
important branch of the Artificial Immune Systems (AIS) [6] [7]. The Ant Colony Optimization
(ACO) algorithm is another emerging approach mimicking the foraging behavior of the ant
species [8].The complex social behaviors of ants have been much studied by science, and
computer scientists are now finding that these behavior patterns can provide models for solving
4 difficult combinatorial optimization problems [3].ACO is the member of swarm intelligence
family and it can constitutes some metaheuristic optimizations. The novel algorithms inspired by
aspect of ant behavior, the ability to find what computer scientists would call shortest paths, has
become the field of ant colony optimization (ACO), the most successful and widely recognized
algorithmic technique based on ant behavior.ACO is a probabilistic technique for solving
computational problems which can be reduced to finding good paths through graph.
Thisalgorithm is initially proposed by Marco Dorigo in 1992 in his PHD thesis [2]. In his thesis
work he presents an overview of this rapidly growing field, from its theoretical inception to
practical applications, including descriptions of many available ACO algorithms and their uses.
In his book ACO is not only introduced but also surveys on ACO applications now in use,
including routing, assignment, scheduling, subset, machine learning, and bioinformatics
problems. AntNet, an ACO algorithm designed for the network routing problem, is described in
detail. On the other hand in EA cross over mutation is very popularmethod of GA. SBX is used
as one point cross over properties for binary GA. It was proposed in 1995 by Deb and Agrawal.
Polynomial is a variant mutation operator in GA. These two are real genetic operation where
fitness function and constrained function are calculated as normal. Researchers use this operator
to propose any hybrid optimization algorithm for variation and efficient results. However, these
powerful optimization methods (ACO) have their inherent shortcomings and limitations. As we
know, fusion and hybrid optimization can give efficient result in some previous case and it
increases efficiency level. Therefore, in this paper, a hybrid algorithm has been introduced using
SBX as crossover operator and Polynomial mutation is performed for mutation. We compare
basic ACO with our hybrid SBXACO by test them upon five benchmark functions.
5 1.1 ACO is member of swarm intelligence
Swarm intelligence is a kind of discipline that deals with the study of self-organizing
processes both in nature and in artificial systems. Recently, computer scientists introduced
algorithms inspired by these models in order to solve difficult various complex
computational problems.ACO is successfully included in swarm intelligence family where
the concept is adopted from folk of ants behaviors. The mathematical function of ACO is
introduced as such concept. The basic ACO is single object Optimization but very recently
MACO is also introduced.
1.2 Motivation
Aco is proposed on 1992, since then it has been use to solve various problems as like as
TSP, bioinformatics, networking, machine learning, variant problem solving and many
other applications. But an ACO method does not always work well and still has room for
improvement on some benchmark functions. This thesis discusses conceptual ACO
algorithm and its modifications. It also describes different types of ACO algorithms and
flowcharts recent works, advanced topic, and application areas of ACO.
1.3 Research Questions
This thesis aims to answer following questions:
Q.1: How functions of ACO parameterized?
Q.2: How efficiently ACO works while tested upon bench mark or test functions?
Q.3: Can we use SBX cross over and Polynomial mutation with ACO?How?
Q.4: How efficiently SBXACO works while tested upon benchmark function?
Q.5: Which one gives best solution? Compare efficiency?
6 CHAPTER 2
Background:
This chapter reviews some of the basic definitions related to this thesis.
2.1 Optimization:
The aim of optimization is to determine the best-suited solution of a problem undera given set of
constraints. For example, in mathematical problem there are several equation where if wide
range taken it become massive problem to choose the best one. Even in industries and scientific
experiment optimization is needed to choose the best way possible. Optimization refer to both
minimization and maximization tasks. Since the maximization of any function is mathematically
equivalent to the minimization of its additive inverse, the term minimization and optimization are
used inter changeably [6]. Optimization problems may be linear or nonlinear.
2.1.1 Constrained Optimization
Many optimization problems require that some of the decision variables satisfy certain
limitations, for instance, all the variables must be non-negative. Such typesof problems are said
to be constrained optimization problems [4] [8] [11] and defined as,
Minimize f(x), x=(x1,x2,x3… xn)
Subject to gm(x)<=0.M=1, 2……ng
hm(x)=0,m=ng+1,……,ng+nh
x Rn
Where ng and nh are the number of inequality constraints respectively.
2.1.2 Unconstrained Optimization
Many optimization problems place no restrictions on the values of that can be assigned to
variables of the problem. The feasible space is simply the whole search space. Such types of
problems are said to be unconstrained optimization problems [4] and defined asminimize
f (x), x Rn. Where n is the dimension of x.
7 2.1.3 Dynamic Optimization
Many optimization problems have objective functions that change over time and such changes in
objective function cause changes in the position of optima. These types of problems are said to
be dynamic optimization problems [4].
2.2.1 Single Object Optimization:
Many real-world decision making problems need to achieve several objectives: minimize risks,
maximize reliability, minimize deviations from desired levels, minimize cost, etc. The main goal
of single-objective (SO) optimization is to find the “best” solution, which corresponds to the
minimum or maximum value of a single objective function that lumps all different objectives
into one. This type of optimization is useful as a tool which should pro-vide decision makers
with insights into the nature of the problem, but usually cannot provide a set of alternative
solutions that trade different objectives against each other.
ABC (Artificial Bee Colony):
The ABC algorithm is proposed by Karaboga [16] in 2005 and the performance of ABC is
analyzed in 2007 [15].The ABC algorithm is developed by inspecting the behaviors of the real
bees on findingfood source, which is called the nectar, and sharing the information of food
sources to the bees in the nest. In the ABC, the artificial agents are defined and classified into
three types, namely, the employed bee, the onlooker bee, and the scout. Each of them plays
different role in the process: the employed bee stays on a food source and provides the
neighborhood of the source in its memory; the onlooker gets the information of food sources
from the employed bees in the hive and select one of the food source to gather thenectar; and the
scout is responsible for finding new food, the new nectar, sources[14].
8 Flow chart of Basic ABC:
Figure 2.1:flow chart of ABC
9 PSO(Particle Swarm Optimization):
The Particle Swarm Optimization algorithm (abbreviated as PSO) is a novel population-based
stochastic search algorithm and an alternative solution to the complex non-linear optimization
problem. The PSO algorithm was first introduced by Dr. Kennedy and Dr. Eberhart in 1995 and
its basic idea was originally inspired by simulation of the social behavior of animals such as bird flocking, fish schooling and so on [10].In PSO, each member of the population is called a particle and the population is called a swarm. Starting with a randomly initialized population and moving in randomly chosen directions, each particle goes through the searching space and remembers the best previous positions of itself and its neighbors. Particles of a swarm communicate good positions to each other as well as dynamically adjust their own position and velocity derived from the best position of all particles. The next step begins when all particles have been moved. Finally, all particles tend to fly towards better and better positions over the searching process until the swarm move to close to an optimum of the fitness function. :
.
Figure 2.2: flow chart ofPSO
10
0 DE(Diifferential Equatio
on):
DE is one
o of the most
m recent EA
E in order to solve reaal-parameteeroptimizatiion problem
ms. Thebasicc
conceptt of DE algoorithm is haaving basicc variant thaat works byy having a ppopulation of
o candidatee
solutionn (called aggents). Theese agents are
a moved around in the search--space by usingsimple
u
e
mathem
matical functtion to com
mbine the poositions of existing agennts from thee populationn. Ifthe new
w
positionn of an agennt is an impprovement itt is acceptedd and formss part of thee population
n, otherwisee
the new
w position iss simply disscarded. Thee process is repeated an
nd by doingg so it is hop
ped, but nott
guarantteed, that a satisfactory
s
y solution will eventually be discovvered. Form
mally, let :
be thee
cost fun
nction whichh must be minimized
m
o fitness funnction whicch must be m
or
maximized [1].
Algoriithm of D
DE:
2.2.2 Multiple
M
O
Object
Op
ptimizatioons:
In our everyday
e
life, we makke decisionss consciously or unconnsciously. T
These decisions can bee
very sim
mple, such as selecting
g the color of
o a dress or
o deciding the menu ffor lunch, orr may be ass
difficultt as those involved in designing a missile orr in selectinng a career.. The formeer decisionss
are easy to make,, while thee latter migght take sevveral yearss due to the level of complexityy
m kinds of
o decision-m
making is too optimize one or morre criteria inn
involveed. The mainn goal of most
order too achieve thhe desired result.
r
In otther words, problems related
r
to ooptimization
n abound inn
real lifee. A multi-oobjective opptimization
n with confl
flicting objeectives, therre is no sin
ngle optimall
solutionn. The interaaction amon
ng differentt objectives gives rise to
t a set of compromiseed solutions,
largely known as thhe trade-offf, non-domiinated, non--inferior or Pareto-optim
P
mal solutionn.
11 2.3.1Evolutionary Algorithm:
Evolutionary algorithm applies the principles of evolution found in nature to the problem of
finding an optimal solution to a solver problem. It is also called genetic algorithm. In a "genetic
algorithm," the problem is encoded in a series of bit strings that are manipulated by the
algorithm; in an "evolutionary algorithm," the decision variables and problem functions are used
directly. Most commercial Solver products are based on evolutionary algorithms.In artificial
intelligence, an evolutionary algorithm (EA) is a subset of evolutionary computation, a generic
population-based Meta heuristic optimizationalgorithm. An EA uses mechanisms inspired
by biological evolution, such as reproduction, mutation, recombination, and selection. Candidate
solutions to the optimization problem play the role of individuals in a population, and the fitness
function determines the quality of the solutions (see also loss function). Evolution of the
population then takes place after the repeated application of the above operators. Artificial
evolution (AE) describes a process involving individual evolutionary algorithms; EAs are
individual components that participate in an AE.In artificial intelligence, an evolutionary
algorithm (EA)
is
a subset of evolutionary
computation,
a
generic
populationbased Metaheuristic optimization algorithm.An evolutionary algorithm for optimization is
different from "classical" optimization methods in several ways:
* Population
*Selection
*CrossOver
*Mutation
2.3.2Population &Selection:
Population where most classical optimization methods maintain a single best solution found so
far, an evolutionary algorithm maintains a population of candidate solutions. Only one (or a few,
with equivalent objectives) of these is "best," but the other members of the population are
"sample points" in other regions of the search space, where a better solution may later be
found.Another term of GA is selection which inspired by the role of natural selection in
evolution – an evolutionary algorithm performs a selection process in which the "most-fit"
12 members of the population survive, and the "least fit" members are eliminated. In a constrained
optimization problem, the notion of "fitness" depends partly on whether a solution is feasible
(i.e. whether it satisfies all of the constraints), and partly on its objective function value. The
selection process is the step that guides the evolutionary algorithm towards ever-better solutions.
2.3.3 Crossover &Mutation:
Crossover inspired by the role of mutation of an organism's DNA in natural evolution -- an evolutionary
algorithm periodically makes random changes or mutations in one or more members of the current
population, yielding a new candidate solution (which may be better or worse than existing population
members).There are many possible ways to perform a "mutation," and the Evolutionary Solver actually
employs three different mutation strategies. The result of a mutation may be an infeasible solution, and
the Evolutionary Solver attempts to "repair" such a solution to make it feasible; this is sometimes, but
not always, successful.Mutation inspired by the role of sexual reproduction in the evolution of living
things -- an evolutionary algorithm attempts to combine elements of existing solutions in order to create
a new solution, with some of the features of each "parent." The elements (e.g. decision variable values)
of existing solutions are combined in a "crossover" operation, inspired by the crossover of DNA strands
that occurs in reproduction of biological organisms.As with mutation, there are many possible ways to
perform a crossover operation -- some much better than others -- and the Evolutionary Solver actually
employs multiple variations of two different crossover strategies.
Simplex Crossover Operator (SPX):
SPX is a multi-parent operator, allowing a user-defined number of parents and offspring. The
parents form a convex hull, called a simplex. Offspring are generated uniformly at random from
within the simplex[18]. The expansion rate parameter can be used to expand the size of the
simplex beyond the bounds of the parents. For example, the figure below shows three parent
points and the offspring distribution, clearly filling an expanded triangular simplex[17].
13 UNDX:
Unimodal Normal Distribution crossover is generates of spring by using the normal distribution
which is defined by µ parent. Offspring are generated around the mean; the probability of
creating an offspring away from the mean vector reduces,preserves the correlation among
parameters well – efficiently solves problem with strong epitasis among parameters, there can be
some areas where it cannot generate offspring from a given initial population, the complexity of
creating one offspring is O(µ2) has difficulties in finding the optimal point(s) near the
boundaries.
PCX:
PCX is known as Parent-Centric Crossover where offspring are centered on each parent; it
assigns more probability for an offspring to remaincloser to the parents than away from parents,
the complexity for creating one offspring is O(µ).
2.4.1: Swarm Intelligence:
Swarm intelligence (SI) is the collective behavior of decentralized, self-organized systems,
natural or artificial[1]. It is a relatively new discipline that deals with the study of self-organizing
processes both in nature and in artificial systems. Researchers in etiology and animal behavior
have proposed many models to explain interesting aspects of social insect behavior such as selforganization and shape-formation. The inspiration often comes from nature, especially biological
systems. Recently, algorithms inspired by these models have been proposed to solve difficult
computational problems [2].This concept was introduced by Gerardo Beni and Jing Wang in
1989, in the context of cellular robotic systems. This expression use widely in artificial
inelegancy. SI systems consist typically of a population of simple agentsor boids interacting
locally with one another and with their environment. The agents follow very simple rules, and
although there is no centralized control structure dictating how individual agents should behave,
local, and to a certain degree random, interactions between such agents lead to the emergenceof
"intelligent" global behavior, unknown to the individual agents. Examples in natural systems of
SI
include ant
colonies,
bird flocking,
animal herding, bacterialgrowth,
fish schooling and microbial intelligence.Some human artifacts also fall into the domain of
swarm intelligence, notably some multi-robot systems, and also certain computer programs that
are written to tackle optimizationand data analysis problems.
14 CHAPTER 3
Basic Ant Colony Optimization:
Ant colony optimization was introduced by M. Dorigo in 1990 for hard combinatorial
Optimization problems [20, 22, and 23]. This Algorithm is used to solve single optimization
problem with reasonable amount of computational time. It is introduced based on the behavior of
real ants. When ants are searching for food they initially explore the area near their nest in a
random manner. If some of them find them find the food source with good amount of food they
came back to the nest with some of the food and they deposit chemical pheromone trail while
they came back. The quantity of pheromone deposits lay on the path they find the food source is
the way of the other ant to find that source easily. This is the indirect way of communication
between the ants via pheromone trails which help them to find the shortest path between the nest
and food source. If the two group of ants find the same sources with two paths, the next ant from
the nest will choose the shortest path from that two path by the justify the amount of pheromone
trails because the amount of pheromone evaporate with time so the shortest path will carry the
more pheromones than the longest path. This characteristic of the real ant colonies are exploited
in artificial ant colonies to solve the optimization problems. In Ant colony optimization
algorithm is parameterized probabilistic model which is called Pheromone model.
Figure: 3.1. ACO model
ACO algorithms are stochastic search procedures. The pheromone model of ACO is the
probabilistic model of search space.
15
5 3.1Thee Basic Model
M
of ACO
A
Algoorithms:
Ant collony optim
mization is a techniquee, which soolve problem
ms byfindinng good paath through
h
graphs. Each ant from
f
the an
nt colony triies to find tthe good paath betweenn its nest an
nd the food
d
sourcess.
16
6 3.1.1 Generate
G
Solution::
This procedure inccludes the routines
r
neeeded for antts to constru
uct solution
ns incremen
ntally. Afterr
the seleection of thee starting citty, one nod
de at a time is added to an ant's patth. An ant's decision off
where to
t go next is biased by
b the pherromone traiils
i,jand
d the heurisstic informaation i,j. In
n
general, the higherr the two values,
v
the higher
h
the probability
p
of choosin
ng the assocciated edge.
Typicallly, two parameters α > 0 and β ≥ 0 are used
u
to weeigh the rellative influeence of thee
pherom
mone and thee heuristic values,
v
respectively. Th
he rule that defines the ant's choicee is specificc
to each ACO variaant.
3.1.2 Daemon
D
A
Action:
This prrocedure comprises alll problem-sp
pecific opeerations thatt may be coonsidered for
f boosting
g
the perfformance off ACO algo
orithms. Thee main exam
mple of such
h operationns is the intrroduction off
a local search phaase. In addiition, daem
mon actions implementt centralizedd tasks thatt cannot bee
perform
med by an in
ndividual ant.
a This typ
pe of procedure is of optional
o
usee, but typically severall
daemon
n actions arre very useeful to signiificantly im
mprove the performancce of ACO algorithmss
(Dorigo
o and Stutzle, 2004).
P
ne Updatee:
3.1.3 Pheromon
Artificial ants moddify some environmen
e
ntal elementts as the real ants. Whhile real antts deposit a
heromone an
nd in artificcial ant colo
ony optimizzation the arrtificial antss
chemicaal substancees called ph
update their pheroomone with
h some num
merical equaations. Pherromone upd
date is very
y importantt
ny optimizaation becau
use If the artificial
a
an
nt competittion the trial then thee
part off Ant colon
mone level will
w update by
b –
pherom
Here
is the am
mount of pheromone
p
deposited in
i respect of
o x, y andd
Evaporaation coeffiicient and of
o course
is the pheromonee
is the amount of pheromone
p
d
deposited
by
y Kth ant.Wee
use this equation to update the pheromone. When this A
Ant colony optimizationn algorithm finds
f
a goodd
value theen the pheromone updatee is needed to
o change thee value with good
g
value.
17 SBXSimulated binary crossover mainly works to simulate the offspring distribution of single-point,
mainly binary encoded crossover on real-valued decision variables. The parameter is generally
for the simulated binary crossover is probabilityand distribution of the index. Here probability
defines the probability of applying the simulatedBinary crossover and distribution Index means
the index of the simulated binary crossoveroperator.Simulated Binary Crossover (SBX) was
proposed in 1995 by Deb and Agrawal.SBX was design with respect to the one-point crossover
properties in binary-coded GA.
Average Property of SBX:
The average property of the decoded parameter values is the same before and after the crossover
operation Spread Factor Property: The probability of occurrence of the spread factor
1 is
more likely than any other value. is defined as the ratio of the spread of offspring point to
that in the parent points :
| c1-c2
1
2)|
Contracting Crossover<1
The offspring points are enclosed by the parent points.
Expanding Crossover>1
The offspring points enclose by the parent points.
Stationary Crossover=1
The offspring points are the same as parent points.
PM:
Polynomial operator attempts to simulate the distribution of binary encoded bit-flip mutation on
the real valued decision variables the polynomial basically favors the offspring nearer to the
parents. The distribution index of polynomial mutation controls the shape of the offspring
18 distribution and the larger values for the distribution index mainly generates offspring closer to
the parents. The operator of the PM is also take two operator like SBX and construct a
polynomial mutation operator with the specified probability and the distribution index.
Flow Chart:
Figure 3.2: Flow chart of SBX and Polynomial
19 3.2 SBX ACO
ACO is a probabilistic technique and search for an optimal path and it’s based on the behavior of
the real ants seeking a path between their colony and source food. Basically to enhance the
efficiency and for a new algorithm SBX-ACO we use simulated binary crossover, polynomial
mutation and ant colony optimization algorithm and try to justify the efficiency level by
benchmark function. We also use polynomial mutation for the better performance of our
algorithm. It’s quite similar to SBX that's why we merge these two operators and include it on
our operation for better performance. As our main target to propose a new algorithm and we
choose ant colony optimization and some other operator like simulated binary crossover and
polynomial mutation because when to research we found that their combination could give good
solution. We choose ACO because the main goal of ant colony optimization is to improve the
performance of algorithm and the second one is to investigate the better explain its behavior.
20
0 Figure: 3..3: Flow chart of SBXA
ACO
21 Pseudo codes:
1. procedure SBX_ACO ()
2. initialize population and parameters
3. determine the parameter of the problem concern
4. create_artificial_ants
5. put the ant on entry level
6.put the other ants on next state
6.while(Iteration is not completed)
8.while(best_indidual_not_found)
7.genetic operation-SBX-crossover
-Mutation
8.Deposit Pheromone
9.Daemon Actions
10.Evaporate pheromone
11.If best individual found
12.end while
13. if best individual not found then again create ants
14.end procedure
22 In SBX-ACO we basically initialize the population and parameters as we require for any
problem and then create artificial ants and put them on the start level of the algorithms the
process will continue till the iteration is not completed our target is to find the best solution from
it so in every time we take the best individuals each time when we get better than it we exchange
it with the previous result. If we did not find any solution then we again start or create ant from
the begging of the algorithm. When we running the algorithmwe run the genetic algorithm
through it we choose SBX-crossover and Mutation before pheromone update. After the
pheromone update we Deposit Pheromone,Daemon Actions Evaporate pheromone as we did in
ACO.Then get the best solution from it.This procedure will be running until we get the solution
3.3Advantage and disadvantage of ACO:
Advantage1. This algorithm is already proven that this algorithm has a positive feedback accounts
for rapid discovery of good solutions
2. This algorithm ensures the interaction of a population agents.
3. ACO is ensures Distributed computation which avoids premature convergence
4. This algorithm is inherent parallelism
5. This is very much efficient for travelling salesman problem
6. This algorithm can be used in dynamic applications
23 Disadvantage1. It is not independent because it’s a sequence of random decisions
2. Probability distribution change due to iteration.
3. Research is experimental rather than theoretical.
4. Convergence is guaranteed but time to convergence is uncertain
3.4 Recent Research and Advanced Topic on ACO:
There are many available successful implantation of ant colony optimization to solve many Meta
heuristic problems. ACO is use to solve Travelling salesman problem. ACO is also applied to
solve many other problems such as vehicle routing problems, scheduling problems. Currently
state-of-the-art for solving the sequential ordering problem, project scheduling problem and open
shop scheduling problem. ACO is also tried to apply on multi-object and the new algorithm for
multi-object is called MACO.Ant colony optimization is use in networking because now a days
it is use to solve in routing problem in telecommunication networks. An improved version of ant
colony optimization is use for solving constraint satisfaction problems. For solving any problems
it can be customized as the requirements and apply ACO can be give a better result. ACO is also
applied in scheduling problems like it is already used in shop scheduling, project scheduling and
able to solve many other scheduling problems with proper solutions. One of the other
applications ACO is to draw Network models.
24 CHAPTER 4
Bench mark Function:
Test function is known as benchmark functions which use vastly in Computer Science. In
applied mathematics, benchmark functions are known as artificial landscapes. These functions
are useful to evaluate characteristics of optimization algorithms according to:
•
•
•
•
Velocity of convergence.
Precision.
Robustness.
General performance
In computer science, benchmark functions are used to check efficiency of optimization
algorithm. The performance evaluates how optimal such algorithm is. It is very useful for
analyzing algorithms upon constraint and unconstraint function. Recently, a paper published by
Noman and Iba evaluate DE algorithm by testing upon them in benchmark Function. They also
introduced a hybrid DEahcSPX and use them solving benchmark functions in order to compare
performances of duo. Basically benchmark functions are mathematical functions but use as a
testing composite in EA. The main reason of using this function is; these mathematical functions
need to be solved by constraint, unconstraint and dynamically optimization where parameter
choosing is very important. Here we choose five benchmark functions with wide range to test
upon. The properties of chosen function are given below.
25
5 Functiion 1:Sph
here
Equattion:
Figurre4.1: Spheree function
Definiition:
•
Search dom
main: −5.12 ≤ xi ≤ 5.12, i = 1, 2. . . .n.
•
ma: no locall minimum except the global one.
Number off local minim
•
The global minima: x*= (0, …, 0)), f(x*) = 0
•
Function grraph: for n = 2
26
6 k
Functiion 2: Rosenbrock
Equattion-
Figure 4.2:
4 Rosenbrrock functionn
Definiition:
•
Number off variables: n variables.
•
main: −5 ≤ xi ≤ 10, i = 1, 2. . . . n.
Search dom
•
ma: several local minim
ma.
Number off local minim
•
f *) = 0.
The global minima: x*= (1… 1), f(x
•
Function grraph: for n = 2.
27
7 Functiion 3: Scaaffer:
Equattion:
Figurre 4.3: Scaffeer function
Definiition:
• The functioon is usuallyy evaluated on the squaare xi [-100, 100], for all i = 1, 2..
•
ff(x*) = 0 x*=(0,0)
•
ma: several local minim
ma.
Number off local minim
•
Function grraph: for n = 2.
28
8 Functiion 4: Grriewank
Equattion:
Figuree 4.4: Griewaank function
Definiition:
•
Number off variables: n variables.
•
Search dom
main: −600 ≤ xi ≤ 600, i = 1, 2, . . . , n.
•
ma: several local minim
mum.
Number off local minim
•
The global minima: x*= (0, …, 0)), f(x*) = 0.
•
Function grraph: for n = 2.
29 Function 5:Ackley
f(x,y)= -20 exp (-0.2 0.5 x
y
– exp (0.5(cos(2πx
Figure 4.5:Ackley function
Definition:
•
Number of variables: n variables.
•
Search domain: −15 ≤ xi ≤ 30, i = 1, 2, . . . , n.
•
Number of local minima: several local minima.
•
The global minimum: x*= (0, …, 0), f(x*) = 0.
•
Function graph: for n = 2.
cos 2πy ))+exp+20
30 CHAPTER 5
Experiments:
Our goal was to make a hybrid of single object optimization algorithm ACO. We use SBX
crossover and proposing a new hybrid named SBXACO. We have carried out different
experiment to assess the performance of SBXACO using the test function described in chapter 4.
Given five test functions are functions commonly found in the literature for CEC 2005 Special
Session on real-parameter optimization [9]. The focus of the study was to compare the
performance of the proposed SBXACO algorithm with the original ACO algorithm in different
experiment.
5.1 Performance evaluation Criteria:
For evaluating the performance of the algorithms, several of the performance criteria of [9] were
used with the difference that 50 instead of 25 trails were conducted, respectively. We compared
the performance of SBXACO with ACO for the test suite using the function error value. The
maximum number of fitness evaluations that we allowed for each algorithm to minimize this
error was 20000*N, where is Nthe dimension of the problem. The fitness evaluation criteria were
as follows.
Evaluation:
The number of function evaluations (FEs) required reaching an error value less than (provided
that recorded in different runs and the average and standard deviation of the number of
evaluations were calculated. For the functions F1 to F2, the accuracy level was fixed at 10-6.For
this criterion, the notation was used AVG SD by which the algorithm could reach the accuracy
level.
5.1.1 Convergence graphs:
Convergence graphs of the algorithms. These graphs show the average FE performance of the
total runs, in respective experiments.
31 5.1.2Experimental Setup:
In our experimentation, we used the same set of initial random populations to evaluate different
algorithms in a similar way done in [1].Though classic ACO uses only three control parameters
namely Update Pheromone, Demon Action and Evaporation. We use SBX operator for crossover
and PM for mutation. For the SBX operation, we chose the number of parents participating in the
0.9 as suggested in [11] and changes to this setting are also
crossover operation to be
examined.The experiments were performed on a computer with 4400 MHz AMD Athlon TM 64
dual core processors and 2 GBof RAM in Java 2 Runtime Environment.
5.1.3Comparison with Basic ACO:
Most of the cases (testing upon function) we get convincing result of our SBXACO rather than
ACO. We could not find better result than ACO only in Ackley function. The table 5.1.1 and
graphfig:(5.1.1-5.1.5) is stated on to show the comparison in below. In graphical comparison it
has shown that ACO performed less efficiently in terms of optimization rather than our proposed
SBXACO.
5.1.4 Comparison with DE and DEachSPX:
In our experiment we take value of DE and DEachSPX from an existing literature publish by
IEEE[1].While comparing our result with their value we found that our hybrid SBX. Some
function were not tested by DE and DEachSPX in the mentioned literature[1] so we ignore those
values ACO shows better performance than DE but DEachSPX works well most of the cases
then us.
Function
Object
Function01
Function02
Function03
Function04
Function05
Minimize
Minimize
Minimize
Minimize
Minimize
Table: 5.1.1
Range
[-5.12,5.12]30
[-2.048,2.048]30
[-100,100]
[[-600,600]10
[-32,32]
32
2 NCTION
FUN
ACO
S
SBX-ACO
hSPX
DEach
DE
15073.6 9461.3
9
60090.0
9499.9(50)
1486550.8 6977.7
(50)
87027.1
1 3967.3
(32)
588987.5
9 (50)
444727.9
535122.9
3661327.2
-
2.Rosennbork
299913.0 519.5
(48)
13559.9
8397.8(50)
51102.7 4325..1
(550)
-
-
fer
3.Scaffe
4.Griennwank
16014.9
6758.4(50)
57202.3
155183.1(50)
1902992.5 63478.8
(40)
121579.2 79563.4
(43)
5.Ackleeys
206306.1
12177.6((50)
2002856.2
10235.5(50)
2154556.1 9721.44
(2)
129211.6 518.6
(45)
1.Spherre
Table: 5.1.2
In
n following graphs FE((Functional Evolution) lies on X-aaxis and Itterartion on Y-axis
Y
F
Figure:
55.1.1
33
3 Here in
n (fig:5.1.1
1)Sphere fu
unction SB
BXACO(Grey) has go
ood result than ACO
O(Red) and
d
DE(yelllow) but DE
EachSPX (B
Blue) which
h perform beetter then alll.
F
Figure:
55.1.2
Here in
n (fig:5.1.2)G
Grienwank function SB
BXACO(Yeellow) has good
g
result than
t
ACO(B
Blue) and
DE(yelllow) but DE
EachSPX (R
Red) which perform beetter then alll.
34
4 F
Figure:
55.1.3
Here in
n (fig:5.1.3)X
X-axis is eq
qual 5000 .R
Rosenbrockk function SBXACO(Y
S
Yellow) has good resultt
than AC
CO(Red) buut DEachSP
PX (Blue) which
w
perforrm better theen all.
F
Figure:
55.1.4
35
5 Here in
n (fig:5.1.4)Scaffer fun
nction SBXA
ACO(Red) has good result
r
than ACO(Blue)
A
) .We could
d
not find
d DE and DE
EachSPX value
v
in men
ntioned[1] literature.
l
F
Figure:
55.1.5
Here in
n (fig:5.1.1)Ackleys function
f
SB
BXACO(Grrey) has go
ood result than
t
ACO((Yellow)butt
DEachS
SPX (Blue) which perfform better then
t
all.
CH
HAPTE
ER 6
Appliccation:
This ch
hapter discusses the varrious appliccation areas of ACO method.This algorithm is a memberr
of the ant colony algorithmss family, in
n swarm intelligence methods,
m
annd it constiitutes somee
Metaheeuristic optim
mizations. Initially
I
pro
oposed by M
Marco Doriigo in 19922 in his PhD
D thesis, thee
first alg
gorithm wass aiming to search for an
a optimal path in a grraph [8], baased on the behavior off
ants seeeking a patth between
n their colony and a source
s
of food.
fo
The ooriginal ideaa has sincee
diversiffied to solvee a wider class of num
merical probblems, and as
a a result, several pro
oblems havee
emerged, drawing on variouss aspects off the behaviior of ants. There are huge
h
field in
i both soft
ft
36 computing to hardware where ACO can use .More over our hybrid SBXACO also can be
implemented for real life problem.
Conceptual design, electromagnetics case,
induction heating cooker design, VLSI
Design and Modeling
design, power systems, RF circuit
synthesis, worst case electronic design,
motor design, filter design, antenna
design, CMOS wideband amplifier
design, logic circuits design, transmission
lines, mechanical design, library search,
inversion of underwater acoustic models,
modeling MIDI music, customer
satisfaction models, thermal process
system identification, friction models,
model selection, ultrawideband channel
modeling, identifying ARMAX models,
power plants and systems, chaotic time
series modeling, model order reduction.
Human tremor analysis for the diagnosis
of Parkinson’s disease, inference of gene
regulatory networks, human movement
biomechanics optimization, RNA
Biomedical
secondary structure determination,
37 phylogenetic tree reconstruction, cancer
classification, and survival prediction,
DNA motif detection, biomarker
selection, protein structure prediction and
docking, drug design, radiotherapy
planning, analysis of brain magneto
encephalography data,
electroencephalogram analysis,
biometrics and so on
Radar networks, bluetooth networks, auto
tuning for universal mobile
telecommunication system networks,
Networking
optimal equipment placement in mobile
communication, TCP network control,
routing, wavelength division-multiplexed
network, peer-to-peer networks,
bandwidth and channel allocation, WDM
telecommunication networks, wireless
networks, grouped and delayed
broadcasting, bandwidth reservation,
transmission network planning, voltage
regulation, network reconfiguration and
expansion, economic dispatch problem,
38 distributed generation,
micro grids, congestion management,
cellular neural networks, design of radial
basis function networks, feed forward
neural network training, product unit
networks, neural gas networks, design of
recurrent neural networks, wavelet neural
networks, neuron controllers, wireless
sensor network design, estimation of
target position in wireless sensor
networks, wireless video sensor networks
optimization
Control of robotic manipulators and arms,
Robotics
motion planning and control, odour
source localization, soccer playing, robot
running, robot vision, collective robotic
search, transport robots, unsupervised
robotic learning, path planning, obstacle
avoidance, swarm robotics, unmanned
vehicle navigation, environment mapping,
voice control of robots, and so forth.
39 Planning landmarks in orthodontic x-ray
images, image classification, inversion of
Image and Graphics
ocean color reflectance measurements,
image fusion, photo time-stamp
recognition, traffic stop-sign detection,
defect detection, image registration,
microwave imaging, pixel classification,
detection of objects, pedestrian detection
and tracking, texture synthesis, scene
matching, contrast enhancement, 3D
recovery with structured beam matrix,
character recognition, image noise
cancellation.
Design of neurofuzzy networks, fuzzy
rule extraction, fuzzy control,
Fuzzy systems, Clustering, data mining
membership functions optimization, fuzzy
modeling, fuzzy classification, design of
hierarchical fuzzy systems, fuzzy queue
management, clustering, clustering in
large spatial databases, document and
information clustering, dynamic
clustering, cascading classifiers,
classification of hierarchical biological
data, dimensionality reduction, genetic-
40 programming-based classification, fuzzy
clustering, classification threshold
optimization, electrical wader sort
classification, data mining, feature
selection
Electrical motors optimization,
optimization of internal combustion
Optimization
engines, optimization of nuclear electric
propulsion systems, floor planning,
travelling-sales man problems, n-queens
problem, packing and knapsack,
minimum spanning trees, satisfiability,
knights cover problem, layout
optimization, path optimization, urban
planning, FPGA placement and routing.
Water quality prediction and
Prediction and forecasting
classification, prediction of chaotic
systems, streamflow forecast, ecological
models, meteorological predictions,
prediction of the floe stress in steel, time
series prediction, electric load forecasting,
battery pack state of charge estimation,
predictions of elephant migrations,
41 prediction of surface roughness in end
milling, urban traffic flow forecasting and
so on.
CHAPTER 7
Conclusion:
This thesis discussed the basic Ant colony Optimization algorithm, geometrical and
mathematical explanation of ACO, demon action, pheromone update, evaporation in the search
space. Function and parameter are also explained in chapter 3.SBX and Polynomial mutation
also described in this chapter. For better explanation flowcharts and pseducode are also
mentioned. Our proposed hybrid SBXACO is also described here with necessary functions.
In chapter 4, our chosen benchmark functions are defined with graph. Here the explanations have
been given that why benchmark function isused to evaluate performances of algorithm.
In chapter 5, Experiments are explained with comparable table and their graph in order to display
the comparisons clearly. Here we state another two algorithm DE and DEachSPX from an
existing literature by Noman and Iba. We use theirresults to compare with our proposed
algorithm.
Basically we try to use GA’s crossover mutation operator and make a hybrid ACO. Due to
justify the efficiency we run it upon some benchmark functions. Later we compare the value with
original ACO which also been tested upon benchmark functions by us.
42 Future plan:
In this paper, we emphasize on single objective ACO. Though we get convincing result we will
submit our paper in journal. Here we work with five benchmark functions,we are working on
test ACO and SBXACO with upon twenty benchmark functions In future we have plan to work
with multi objective ACO also. Moreover PSO is another single objective optimization algorithm
which actually can be use along with ACO to introduce a hybrid. We are eager to work with
PSO. Moreover, this thesis can be used in various applications like networking, graphics,
bioinformatics, robotics,design and modeling etc. We are also eager to implement our algorithm
in an application in future.
References:
[1]N.Nasimul and H.Iba Accelerating Differential Evolution Using Adaptive Local Search,IEEE
vol.12,no:01,pp 107-124,2008.
[2]Patel,K.,Kabat,R and Tripathy.2013.A hybrid ACO/PSO based algorithm for QoS multicast
routing problem,inIndia,Dept. of Comp. Sc. &Engg.,Veer SurendraSai University of
Technology;inAin Shams Engineering Journal,Ain Shams University:113-120.
[3]T.Satyobroto,”Mathmatical
Modelling
and
Application
of
Optimization”,School of Engineering,Blekinge Institute of Technology;.
Particle
Swarm
[4] Andries P. Engelbrecht, Computational Intelligence: An Introduction.: John
Wiley and Sons, 2007, ch. 16, pp. 289-358
[5] Singiresu S. Rao, Engineering Optimization Theory and Practice, 4th edition, Ed.: John
Wiley and Sons, 2009.
[6]Singiresu S.Rao,Engineering Optimisation Theory And Practiceth,4th edition,Ed:John Wiley
and Sons,2009
[7]El-Ghazali Talbi,Metahurestic-Form Design to Implementation::John Wiley and Sons,2009.
[8]Dorigo, M., Birattari, M., and Stutzle, T., Ant colony optimization, IEEE Computational
Intelligence .Magazine, 1(4):28-39, 2006.
[9]P. N. Suganthan, N. Hansen, J. J. Liang, K. Deb, Y.-P. Chen, A. Auger,and S. Tiwari,
43 “Problem Definitions and Evaluation Criteria for theCEC 2005 Special Session on RealParameter Optimization,” NanyangTechnol. Univ., Singapore, IIT Kanpur, India, KanGAL Rep.
2005005,May 2005
[10]Noman, Iba and Hitoshi ;Member, IEEE.2008 . Accelerating Differential Evolution
Using an Adaptive Local Search; in IEEE TRANSACTIONS ON EVOLUTIONARY
COMPUTATION,12(1):108-124.
[11] S. Tsutsui, M. Yamamura, and T. Higuchi, “Multi-parent recombination with simplex
crossover in real coded genetic algorithms,” in Proc.Genetic Evol. Comput. Conf. (GECCO’99),
Jul. 1999, pp. 657–664.
[12] Mukharjee.D and Acharyya.S.Ant Colony Technique applied in Network Routing Problem,
in West Bengal University of Technology;in International journal of Computer Applications
1(15).
[13] Suganthan,P.N,Hasen.N,Liang,J.J,Deb.K,Auger.A and Tiwari.S,2005.Problem Difinatiions
and Evaluation Criteria for the CEC Special Session on Real-Parameter Optimization.Nayanyang
Technological University,Singapore,School of EEE.
[14]C. M. Bishop. Neural Networks for Pattern Recognition. Oxford University Press,
1995.
[15] J. Branke, M. Middendorf, and F. Schneider. Improved heuristics and a genetic
algorithm for finding short supersequences. OR-Spektrum, 20:39–46, 1998.
[14]P.W .Tsai, J.S.Pan, B. Y. Liao and S.C.Chu,Enhanced Artificial Bee Colony
Optimization,ICIC
International,ISSN 1349-4198,Cheng Shiu University,2009.
[16] D. Karaboga, An Idea Based On Honey Bee Swarm For Numerical Optimization, Technical
ReportTR06, Erciyes University, Engineering Faculty, Computer Engineering Department, 2005.
[17] K. W. Kim, M. Gen, and M. Kim, Adaptive Genetic Algorithms for Multi-resource
Constrained
Project Scheduling Problem with Multiple Modes, International Journal of Innovative
Computing,
Information and Control, vol.2, no.1, pp.41-49, 2006.
[18]Tsutsui, S., Yamamura, M., and Higuchi, T., "Multi-parent Recombination with Simplex
Crossover in Real Coded Genetic Algorithms," Proceedings of the Genetic and Evolutionary
Computation Conference, vol. 1, pp. 657-664, 1999.
44 [19]Higuchi, T., Tsutsui, S., and Yamamura, M., "Theoretical Analysis of Simplex Crossover for
Real-Coded Genetic Algorithms," Parallel Problem Solving from Nature PPSN VI, pp. 365-374,
2000.
[20]Savic.D, “Single – Objective vs Multiobjective optimization for Integrate Decision
Support”.University of Exeter,UK.Department of Engineering School and ComputerScience
[21]Bandyopadhaye,S;
Saha,S;”
Applications,pp.262,2013.
”.Classical
and
Metahurestic
Approaches
and
[22] M. Fatih Tasgetiren and Yun-Chia Liang, "A Binary Particle Swarm
Optimization Algorithm for Lot Sizing Problem," Journal of Economic and
Social Research, vol. 5, no. 2, pp. 1-20, 2003.
[23] D. Corne, M. Dorigo, and F. Glover, editors. New Ideas in Optimization. McGrawHill,
1999.
[24] D. Costa and A. Hertz. Ants can colour graphs. Journal of the Operational Research
Society, 48:295–305, 1997.
[25] D. Costa, A. Hertz, and O. Dubuis. Embedding of a sequential algorithm within
an evolutionary algorithm for coloring problems in graphs. Journal of Heuristics,
1:105–128, 1995.
[26] G.A. Croes. A method for solving traveling salesman problems. Operations Research,
6:791–812, 1958.
[27] J.-L. Deneubourg, S. Aron, S. Goss, and J.-M. Pasteels. The self-organizing exploratory
pattern of the argentine ant. Journal of Insect Behavior, 3:159–168, 1990.
[28] G. Di Caro and M. Dorigo. AntNet: A mobile agents approach to adaptive routing.
Technical Report 97-12, IRIDIA, Universit´e Libre de Bruxelles, 1997.
[29] G. Di Caro and M. Dorigo. Mobile agents for adaptive routing. In Proceedings of
the 31st International Conference on System Sciences (HICSS-31), volume 7, pages
74–83. IEEE Computer Society Press, 1998.
[30] Muhlenbein, R. , Schom isch, M., and Born, J. , "T he Par allel Genetic Algorit
hms as Function Optimizer, " in Proceedings of the FOU1,th International
Conference on Genetic Algorithms, edited by R. K, Belew and L. B. Booker
(Morgan Kaufmann, San Mateo, 1991).
[31] Gordon, V. S. and Whitley, D., "Serial and Par allel Genetic Algorithms as
Function Optimizers ," in Proceedings of the Fifth International Conference
on Genetic Algorithms, edite d by S. Forrest (Mor gan Kaufmann, San Mat eo
1993) .
45 [32] Deb , K. , "Genetic Algorithms in Multimodal Funct ion Optimization," TCGA
Repo rt No. 89002 (Uni versity of Alabama , Tus caloosa , 1989).
[33] Gold berg , D. E., an d Richardson , J ., "Genet ic Algori thms with Sharing for
Multimod al Function Optimizati on ," in Proceedings of the Second International
Confe rence on Geneti c Algor-ithms, edited by J. J. Grefenste tte (Morgan
Kaufmann, San Mateo, 1987).
[34] Srinivas , N. an d Deb , K. , "Multiobj ecti ve Function Optimization using Nondominate
d Sorting in Genet ic Algorit hms," Evolutionary Computation, 2( 3) ,
1995, 221-248.